Six factor formula explained

The six-factor formula is used in nuclear engineering to determine the multiplication of a nuclear chain reaction in a non-infinite medium.

Six-factor formula: k = η f p \varepsilon P _{FNL} P _{TNL} = k _{infty} P _{FNL} P _{TNL} ^[1] ! Symbol! Name! Meaning! Formula! Typical thermal reactor value

η	Thermal fission factor (eta)		η = (\nu \sigma ^{F} _{f})/(\sigma ^{F} _{a}) = (\nu \Sigma ^{F} _{f})/(\Sigma ^{F} _{a})	1.65
f	Thermal utilization factor		f = (\Sigma ^{F} _{a})/(\Sigma _{a})	0.71
p	Resonance escape probability		p ≈ exp \left( - (\sum\limits ^{N} _{i =1} N _{i} I _{r,A,i})/(\left( \overline{\xi) \Sigma _{p} \right) _{mod} } \right)	0.87
\varepsilon	Fast fission factor (epsilon)		\varepsilon ≈ 1 + (1- p)/(p) (u _{f} \nu _{f} P _{FAF})/(f \nu _{t} P _{TAF} P _{TNL})	1.02
P _{FNL}	Fast non-leakage probability		P _{FNL} ≈ exp \left( - {B ^{2} _{g}} \tau _{th} \right)	0.97
P _{TNL}	Thermal non-leakage probability		P _{TNL} ≈ (1)/(1+ {L _{th}) ^{2} {B ^{2 }} _{g}}	0.99

The symbols are defined as:^[2]

\nu , \nu _{f} and \nu _{t} are the average number of neutrons produced per fission in the medium (2.43 for uranium-235).\sigma ^{F} _{f} and \sigma ^{F} _{a} are the microscopic fission and absorption cross sections for fuel, respectively.\Sigma ^{F} _{a} and \Sigma _{a} are the macroscopic absorption cross sections in fuel and in total, respectively.\Sigma ^{F} _{f} is the macroscopic fission cross-section.N _{i} is the number density of atoms of a specific nuclide.I _{r,A,i} is the resonance integral for absorption of a specific nuclide.I _{r,A,i} = \int ^{E _{0}} _{E _{th}} dE' (\Sigma ^{mod} _{p})/(\Sigma _{t(E')}) (\sigma ^{i(E')} _{a})/(E') \overline{\xi} is the average lethargy gain per scattering event.

• • Lethargy is defined as decrease in neutron energy.

u _{f} (fast utilization) is the probability that a fast neutron is absorbed in fuel.P _{FAF} is the probability that a fast neutron absorption in fuel causes fission.P _{TAF} is the probability that a thermal neutron absorption in fuel causes fission.{B ^{2} _{g}} is the geometric buckling.{L _{th} }^2 is the diffusion length of thermal neutrons.{L _{th} }^2 = \frac\tau _{th} is the age to thermal.\tau = \int ^{E'} _{E _{th}} dE'' (1)/(E'') (D(E''))/(\overline{\xi) \left[ D(E'') {B ^{2} _{g}} + \Sigma _{t(E')} \right]} \tau _{th} is the evaluation of \tau where E' is the energy of the neutron at birth.

Multiplication

The multiplication factor,, is defined as (see nuclear chain reaction):

• If is greater than 1, the chain reaction is supercritical, and the neutron population will grow exponentially.
• If is less than 1, the chain reaction is subcritical, and the neutron population will exponentially decay.
• If, the chain reaction is critical and the neutron population will remain constant.

See also

• Critical mass
• Nuclear chain reaction
• Nuclear reactor
• Four factor formula

Notes and References

1. Book: Duderstadt, James . Hamilton, Louis . Nuclear Reactor Analysis . 1976 . John Wiley & Sons, Inc . 0-471-22363-8 .
2. Book: Adams, Marvin L. . Introduction to Nuclear Reactor Theory . 2009 . Texas A&M University.